What you have is a not for profit rental.

Go to IRS Publication 527 Residential Rental Property page 16 - https://www.irs.gov/pub/irs-pdf/p527.pdf#page=16

If you don’t rent your property to make a profit, you can’t deduct rental expenses in excess of the amount of your rental income. You can’t deduct a loss or carry forward to the next year any rental expenses that are more than your rental income for the year.
Where to report. Report your not-for-profit rental income on Schedule 1 (Form 1040), line 8. If you itemize your deductions, include your mortgage interest and mortgage insurance premiums (if you use the property as your main home or second home), real estate taxes, and casualty losses from your not-for-profit rental activity when figuring the amount you can deduct on Schedule A.

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@VTer2021 wrote:

This will result in our not using this second home for any tax deductions. 

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Note that you can claim a deduction for the real estate tax. That's included in the list of Schedule A itemized deductions at the end of DoninGA's reply above. Of course, this is subject to the $10,000 cap on the total deduction for state and local taxes.

We will not be itemizing any deductions on our tax return (we use the stand deduction) and I'm trying to avoid having to put anything related to this house on our return.  That is why I was wondering if we don't itemize and are not planning to deduct anything related to this house on our return if we needed to include her payments for the utilities as "rent".  If we had to file a rental income schedule, her payments would just zero any expenses relating to the house.  This is not a rental property...just a place for her to stay for a while until it becomes our primary residence.

We will not be itemizing any deductions on our tax return (we use the standard deduction) and I'm trying to avoid having to put anything related to this house on our return.  That is why I was wondering if we don't itemize and are not planning to deduct anything related to this house on our return if we needed to include her payments for the utilities as "rent".  If we had to file a rental income schedule, her payments would just zero any expenses relating to the house.  This is not a rental property...just a place for her to stay for a while until it becomes our primary residence.

You do not have a  "not for profit rental", in my opinion.  You have a second home with a family member "sharing" expenses.  You do not have to report those expenses as rental income, on your tax return. 

@Hal_Al is right.....this is a cost sharing arrangement and is nowhere near any type of rental where income would be reported on SCH E or as misc income. No need to report.